Large Aperiodic Semigroups

Abstract

Abstract. The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of a minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having n left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with n states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For n 4 there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that 2 n − 1 is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.

Syntactic Complexity of Bifix-Free Languages

“…For example, the state complexity of basic operations in the class of star-free languages is the same as in the class of all regular languages (except the reversal, where the tight upper bound is 2 n−1 − 1 see [8]). Finally, the largest transition semigroups play an important role in the study of most complex languages [3] in a given subclass.…”

“…Syntactic complexity has been studied for a number of subclasses of regular languages (e.g., [4,5,6,8,13,14]). For bifix-free languages, the lower bound (n − 1) n−3 + (n − 2) n−3 + (n − 3)2 n−3 for the syntactic complexity for n 6 was established in [6].…”

Syntactic Complexity of Regular Ideals

“…The problem we study in this paper is the following: Given a language belonging to a subclass of the class of regular languages -for example, the subclass of finite languages or prefix-free languages (prefix-codes) -what is the maximal size of the Syntactic complexity has been studied in several subclasses of regular languages other than ideals: prefix-, suffix-, bifix-, and factor-free languages [8,12]; star-free languages [7,10]; R-and J -trivial languages [6]; finite/cofinite and reverse definite languages [7]. This problem can be quite challenging, depending on the subclass; in the present case it is easy for right ideals but much more difficult for left-and two-sided ideals (defined below).…”

“…Holzer and König [19], and independently Krawetz, Lawrence, and Shallit [20] studied the syntactic complexity of unary and binary regular languages. Recently, syntactic complexity has been studied in several subclasses of regular languages other than ideals: prefix-, suffix-, bifix-, and factor-free languages [8,12]; star-free languages [7,10]; R-and J -trivial languages [6].…”